About the Project
18 Orthogonal PolynomialsGeneral Orthogonal Polynomials

§18.2 General Orthogonal Polynomials

  1. §18.2(i) Definition
  2. §18.2(ii) x-Difference Operators
  3. §18.2(iii) Normalization
  4. §18.2(iv) Recurrence Relations
  5. §18.2(v) Christoffel–Darboux Formula
  6. §18.2(vi) Zeros

§18.2(i) Definition

Orthogonality on Intervals

Let (a,b) be a finite or infinite open interval in . A system (or set) of polynomials {pn(x)}, n=0,1,2,, is said to be orthogonal on (a,b) with respect to the weight function w(x) (0) if

18.2.1 abpn(x)pm(x)w(x)dx=0,

Here w(x) is continuous or piecewise continuous or integrable, and such that 0<abx2nw(x)dx< for all n.

It is assumed throughout this chapter that for each polynomial pn(x) that is orthogonal on an open interval (a,b) the variable x is confined to the closure of (a,b) unless indicated otherwise. (However, under appropriate conditions almost all equations given in the chapter can be continued analytically to various complex values of the variables.)

Orthogonality on Finite Point Sets

Let X be a finite set of distinct points on , or a countable infinite set of distinct points on , and wx, xX, be a set of positive constants. Then a system of polynomials {pn(x)}, n=0,1,2,, is said to be orthogonal on X with respect to the weights wx if

18.2.2 xXpn(x)pm(x)wx=0,

when X is infinite, or

18.2.3 xXpn(x)pm(x)wx=0,

when X is a finite set of N+1 distinct points. In the former case we also require

18.2.4 xXx2nwx<,

whereas in the latter case the system {pn(x)} is finite: n=0,1,,N.

More generally than (18.2.1)–(18.2.3), w(x)dx may be replaced in (18.2.1) by a positive measure dα(x), where α(x) is a bounded nondecreasing function on the closure of (a,b) with an infinite number of points of increase, and such that 0<abx2ndα(x)< for all n. See McDonald and Weiss (1999, Chapters 3, 4) and Szegő (1975, §1.4).

§18.2(ii) x-Difference Operators

If the orthogonality discrete set X is {0,1,,N} or {0,1,2,}, then the role of the differentiation operator d/dx in the case of classical OP’s (§18.3) is played by Δx, the forward-difference operator, or by x, the backward-difference operator; compare §18.1(i). This happens, for example, with the Hahn class OP’s (§18.20(i)).

If the orthogonality interval is (,) or (0,), then the role of d/dx can be played by δx, the central-difference operator in the imaginary direction (§18.1(i)). This happens, for example, with the continuous Hahn polynomials and Meixner–Pollaczek polynomials (§18.20(i)).

§18.2(iii) Normalization

The orthogonality relations (18.2.1)–(18.2.3) each determine the polynomials pn(x) uniquely up to constant factors, which may be fixed by suitable normalization.

If we define

18.2.5 hn=ab(pn(x))2w(x)dx or xX(pn(x))2wx,
18.2.6 h~n=abx(pn(x))2w(x)dx or xXx(pn(x))2wx,


18.2.7 pn(x)=knxn+k~nxn1+k~~nxn2+,

then two special normalizations are: (i) orthonormal OP’s: hn=1, kn>0; (ii) monic OP’s: kn=1.

§18.2(iv) Recurrence Relations

As in §18.1(i) we assume that p1(x)0.

First Form

18.2.8 pn+1(x)=(Anx+Bn)pn(x)Cnpn1(x),

Here An, Bn (n0), and Cn (n1) are real constants, and An1AnCn>0 for n1. Then

18.2.9 An =kn+1kn,
Bn =(k~n+1kn+1k~nkn)An=h~nhnAn,
Cn =Ank~~n+Bnk~nk~~n+1kn1=AnAn1hnhn1.

Second Form

18.2.10 xpn(x)=anpn+1(x)+bnpn(x)+cnpn1(x),

Here an, bn (n0), cn (n1) are real constants, and an1cn>0 (n1). Then

18.2.11 an =knkn+1,
bn =k~nknk~n+1kn+1=h~nhn,
cn =k~~nank~~n+1bnk~nkn1=an1hnhn1.

If the OP’s are orthonormal, then cn=an1 (n1). If the OP’s are monic, then an=1 (n0).

Conversely, if a system of polynomials {pn(x)} satisfies (18.2.10) with an1cn>0 (n1), then {pn(x)} is orthogonal with respect to some positive measure on (Favard’s theorem). The measure is not necessarily of the form w(x)dx nor is it necessarily unique.

§18.2(v) Christoffel–Darboux Formula

18.2.12 =0np(x)p(y)h=knhnkn+1pn+1(x)pn(y)pn(x)pn+1(y)xy,

Confluent Form

18.2.13 =0n(p(x))2h=knhnkn+1(pn+1(x)pn(x)pn(x)pn+1(x)).

§18.2(vi) Zeros

All n zeros of an OP pn(x) are simple, and they are located in the interval of orthogonality (a,b). The zeros of pn(x) and pn+1(x) separate each other, and if m<n then between any two zeros of pm(x) there is at least one zero of pn(x).

For illustrations of these properties see Figures